Calculus of Variation

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1、CALCULUSOFVARIATIONSMA4311LECTURENOTESI.B.RussakDepartmentofMathematicsNavalPostgraduateSchoolCodeMA/RuMonterey,California93943July9,2002
c1996-ProfessorI.B.Russak1Contents1FunctionsofnVariables11.1UnconstrainedMinimum.............................11.2

2、ConstrainedMinimization............................52Examples,Notation102.1Notation&Conventions.............................132.2ShortestDistances................................143FirstResults213.1TwoImportantAuxiliaryFormulas:.......................

3、223.2TwoImportantAuxiliaryFormulasintheGeneralCase............264VariableEnd-PointProblems364.1TheGeneralProblem...............................384.2Appendix.....................................415HigherDimensionalProblemsandAnotherProofoftheSecondEule

4、rEquation465.1VariationalProblemswithConstraints.....................475.1.1IsoparametricProblems..........................475.1.2PointConstraints..............................516IntegralsInvolvingMoreThanOneIndependentVariable597ExamplesofNumericalTe

5、chniques637.1IndirectMethods.................................637.1.1FixedEndPoints..............................637.1.2VariableEndPoints............................717.2DirectMethods..................................748TheRayleigh-RitzMethod828.1Euler

6、’sMethodofFiniteDifferences.......................849Hamilton’sPrinciple9010DegreesofFreedom-GeneralizedCoordinates9711IntegralsInvolvingHigherDerivatives10412PiecewiseSmoothArcsandAdditionalResults11013FieldTheoryJacobi’sNeccesaryConditionandSufficiency

7、116iListofFigures1NeighborhoodSofX0..............................22NeighborhoodSofX0andaparticulardirectionH..............23TwodimensionalneighborhoodofX0showingtangentatthatpoint.....54Theconstraintφ.................................65Thesurfaceofrevo

8、lutionforthesoapexample.................116Brachistochroneproblem.............................127AnarcconnectingX1andX2..........................158Admissiblefunctionηvanishingatendpoints(bottom)andvariousadmissiblefunctions(top)..............